Interpolation of Fredholm operators

TL;DR

This paper establishes new conditions under which Fredholm operators maintain their properties under real interpolation, providing a comprehensive framework that includes an abstract factorization theorem and applications to spectral theory.

Contribution

It introduces sufficient and necessary conditions for Fredholm operators to be preserved under real interpolation, including an abstract factorization theorem and solutions to the Lions-Magenes problem.

Findings

Provided conditions for Fredholm operators on interpolated spaces.

Solved the Lions-Magenes problem on real interpolation of subspaces.

Discussed applications to spectral theory and Hardy operator perturbations.

Abstract

We prove novel results on interpolation of Fredholm operators including an abstract factorization theorem. The main result of this paper provides sufficient conditions on the parameters $\theta \in (0,1)$ and $q\in \lbrack 1,\infty ]$ under which an operator $A$ is a Fredholm operator from the real interpolation space $(X_{0},X_{1})_{\theta ,q}$ to $(Y_{0},Y_{1})_{\theta ,q} $ for a given operator $A\colon (X_{0},X_{1})\rightarrow (Y_{0},Y_{1})$ between compatible pairs of Banach spaces such that its restrictions to the endpoint spaces are Fredholm operators. These conditions are expressed in terms of the corresponding indices generated by the $K$-functional of elements from the kernel of the operator $A$ in the interpolation sum $X_{0}+X_{1}$. If in addition $q\in \lbrack 1,\infty )$ and $A$ is invertible operator on endpoint spaces, then these conditions are also necessary. We apply these results to obtain and present an affirmative solution of the famous Lions-Magenes problem on the real interpolation of closed subspaces. We also discuss some applications to the spectral theory of operators as well as to perturbation of the Hardy operator by identity on weighted $L_{p}$-spaces.